Properties

Degree 2
Conductor $ 2^{2} \cdot 3 \cdot 5^{2} \cdot 7 $
Sign $0.447 + 0.894i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + i·3-s i·7-s − 9-s − 6·11-s + 2i·13-s + 4·19-s + 21-s − 6i·23-s i·27-s − 6·29-s + 8·31-s − 6i·33-s − 2i·37-s − 2·39-s + 12·41-s + ⋯
L(s)  = 1  + 0.577i·3-s − 0.377i·7-s − 0.333·9-s − 1.80·11-s + 0.554i·13-s + 0.917·19-s + 0.218·21-s − 1.25i·23-s − 0.192i·27-s − 1.11·29-s + 1.43·31-s − 1.04i·33-s − 0.328i·37-s − 0.320·39-s + 1.87·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(2100\)    =    \(2^{2} \cdot 3 \cdot 5^{2} \cdot 7\)
\( \varepsilon \)  =  $0.447 + 0.894i$
motivic weight  =  \(1\)
character  :  $\chi_{2100} (1849, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 2100,\ (\ :1/2),\ 0.447 + 0.894i)\)
\(L(1)\)  \(\approx\)  \(1.068786012\)
\(L(\frac12)\)  \(\approx\)  \(1.068786012\)
\(L(\frac{3}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;5,\;7\}$,\(F_p(T)\) is a polynomial of degree 2. If $p \in \{2,\;3,\;5,\;7\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 - iT \)
5 \( 1 \)
7 \( 1 + iT \)
good11 \( 1 + 6T + 11T^{2} \)
13 \( 1 - 2iT - 13T^{2} \)
17 \( 1 - 17T^{2} \)
19 \( 1 - 4T + 19T^{2} \)
23 \( 1 + 6iT - 23T^{2} \)
29 \( 1 + 6T + 29T^{2} \)
31 \( 1 - 8T + 31T^{2} \)
37 \( 1 + 2iT - 37T^{2} \)
41 \( 1 - 12T + 41T^{2} \)
43 \( 1 + 4iT - 43T^{2} \)
47 \( 1 + 12iT - 47T^{2} \)
53 \( 1 + 6iT - 53T^{2} \)
59 \( 1 + 59T^{2} \)
61 \( 1 + 10T + 61T^{2} \)
67 \( 1 + 8iT - 67T^{2} \)
71 \( 1 - 6T + 71T^{2} \)
73 \( 1 + 10iT - 73T^{2} \)
79 \( 1 - 4T + 79T^{2} \)
83 \( 1 + 12iT - 83T^{2} \)
89 \( 1 + 12T + 89T^{2} \)
97 \( 1 - 10iT - 97T^{2} \)
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\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−9.056382081793747333227466982444, −8.112979180081501232426812543219, −7.58239115349956445927609861427, −6.62069975807823456338094688079, −5.61235911729251425840469039932, −4.94175526113973912105672302566, −4.13766361371917066456946450974, −3.07919939410614854098454602045, −2.20418094039544649709065316617, −0.40506453482796761740976323754, 1.16120108529434125444805543616, 2.56113527066698518566206620410, 3.07240320708764163054184539503, 4.50078541102630895590818772348, 5.56894161311508485763590873490, 5.79532631877383755956047939166, 7.09959069637711725499202161259, 7.84588758143579943271053413032, 8.096362834242939938755733891477, 9.334160588088703665723149310247

Graph of the $Z$-function along the critical line